Modifications of the Page Curve from correlations within Hawking radiation
Mishkat Al Alvi, Mahbub Majumdar, Md. Abdul Matin, Moinul Hossain Rahat, Avik Roy
MModifications of the Page Curve from Correlations withinHawking Radiation
Mishkat Al Alvi, ∗ Mahbub Majumdar, † Md. Abdul Matin(deceased), Moinul Hossain Rahat, ‡ and Avik Roy § Department of Physics and Astronomy,University of Minnesota, Duluth, Minnesota 55812, USA BRAC University, 66 Mohakhali, Dhaka 1212, Bangladesh Bangladesh University of Engineering & Technology, Dhaka 1000, Bangladesh Institute for Fundamental Theory, Department of Physics,University of Florida, Gainesville, Florida 32611, USA Center for Particles and Fields, Department of Physics,University of Texas at Austin, Austin, Texas 78712, USA
Abstract
We investigate quantum correlations between successive steps of black hole evaporation andinvestigate whether they might resolve the black hole information paradox. ‘Small’ correctionsin various models were shown to be unable to restore unitarity. We study a toy qubit modelof evaporation that allows small quantum correlations between successive steps and reaffirmprevious results. Then, we relax the ‘smallness’ condition and find a nontrivial upper andlower bound on the entanglement entropy change during the evaporation process. This gives aquantitative measure of the size of the correction needed to restore unitarity. We find that theseentanglement entropy bounds lead to a significant deviation from the expected Page curve.
Keywords: Black hole information paradox, Toy Model Black Hole, Qubit Model black hole, Quantumcorrelations, Small correction ∗ [email protected] † [email protected], [email protected] ‡ mrahat@ufl.edu § [email protected] a r X i v : . [ h e p - t h ] A ug . INTRODUCTION In 1972 Bekenstein audaciously associated a thermodynamic entropy to black holesthat was proportional to the horizon area [1, 2]. This picture was elaborated in [3–5],and a complete description of classical black hole mechanics was presented in [5, 6].The conflict between classical gravity that asserted that black holes don’t emit radi-ation, and the thermodynamic picture whereby they have a temperature, was resolvedby Hawking in 1975 [7]. Using what are now standard techniques in curved space quan-tum field theory, Hawking showed that a black hole radiates as a black body with atemperature of κ π .Hawking suggested a heuristic picture whereby pair production occurred around thehorizon. One particle fell into the black hole and another particle – the Hawking radiation– escaped to asymtopia. The picture was given a concrete realization in terms of tunnelingby Parikh and Wilczek [8].However, this produced a new paradox. The ingoing and outgoing Hawking radiationfrom pair production would be entangled. Thus, as the black hole evaporated, the entan-glement entropy of the outgoing radiation would steadily increase. The emitted Hawkingradiation at the end of the evaporation would then be entangled with “nothing.” Thusa black hole that began in a pure state would end in a mixed state violating unitarity.This is the black hole information paradox. Suppose it were possible to transfer the entanglement between the outgoing radiationand radiation/matter inside the black hole, to the outgoing radiation that came out late.Then all of the information in the black hole could be carried out by late time Hawkingradiation and the final state would be a pure state – a pure state of early outgoingradiation entangled with late outgoing radiation.It was believed that one mechanism to realize this picture are ‘small’ correlationsbetween the Hawking quanta. Many ‘small’ effects might conceivably, collectively add upand allow for all of the information to come out [9]. Using strong subadditivity, Mathurshowed that in a simple model of Hawking pairs being Bell pairs, that small correlationswere unable to decrease the entanglement entropy enough to preserve unitarity [10].Later, Mathur showed that small correlations between consecutive, local, Hawking pairemissions – such that a Hawking pair was correlated to the next emitted Hawking pair2 was likewise unable to reduce the entanglement entropy of the Hawking radiation tozero [11]. In [12], three models of non-local correlations among the Hawking radiationwere considered and shown to not appreciably decrease the outgoing Hawking radiationentanglement entropy. Giddings in [13] presented a nonlocal qubit model that was unitary.In [14], Avery gave a framework to describe both unitary and non-unitary models of blackhole evaporation and gave an example of how a non-unitary model could through a largedeformation be made to be unitary.In order to ensure that the horizon has no drama , it has been argued that correctionsto Hawking pairs should be small. In the presence of small corrections, the niceness conditions would still hold at the horizon which would enable the equivalence principleto hold at the horizon. Large O (1) Hawking pair corrections were argued to be able torestore unitarity, but would remove the no drama nature of the horizon [10, 11, 15, 16].In this paper, we continue the study of small correlations in restoring unitarity. Weprovide a more general analysis than [10]. We allow arbitrary, small correlations betweenHawking pairs. We show that such corrections do not restore unitarity. We next relax the‘smallness’ condition as prescribed in [10] and find a nontrivial upper and lower boundon the entanglement entropy change. This formulation allows us to quantify the kind ofcorrections required to restore unitarity.We find that our nontrivial bounds on the entanglement entropy give a deviation fromthe expected Page Curve [17–19]. This leads us to conclude that the corrections dictatedby our qubit formalism are not compatible with physical unitary evolution. This resultis in line with a more general result earlier proved in [13] – that corrections in the formof admixtures of Bell pair states alone, are insufficient to restore unitarity.The organization of the paper is as follows. Section II introduces the leading orderformulation of the black hole information paradox. In section III, we describe the frame-work often used to try to resolve the paradox using ‘small’ corrections. In section IV, asimple toy model supporting the results of the previous section is briefly analyzed andshown to be unable to halt the monotonic growth of entanglement entropy. Section Vgeneralizes the arguments in section III and finds upper and lower bounds on the changeof entanglement entropy ∆ S during the evaporation process when large correlations be-tween Hawking quanta are allowed. Section VI shows how the upper and lower boundson ∆ S deduced in Section V produce a modified Page Curve. The modified Page Curve3ints that Bell pair corrections are insufficient to make the evaporation process unitary.Section VII summarizes our findings. II. LEADING ORDER FORMULATION OF THE BLACK HOLE INFORMA-TION PARADOX
In this section we review the formulation of the information paradox as presented in[10] and that was followed by others. The purpose is to familiarize the reader with theframework we will use.We make certain assumptions that will allow us to use local effective field theory tohandle the quantum gravity effects that produce Hawking radiation. This means thatgiven a quantum state on a spacelike slice, we can evolve forward to future spacelike slicesusing Hamiltonian evolution. The assumptions are:1. The black hole geometry can be foliated by spacelike slices continuous at the horizonand the physics on these slices is given by a local quantum field theory [10, 20].2. Suppose a collapsing shell with state | Ψ (cid:105) M produces the black hole. Time evolutionof the hole, will push this matter far along spacelike slices inside the hole such thatit is far from where Hawking radiation is being produced. Thus | Ψ (cid:105) M will at most,weakly affect the Hawking pairs. Thus, after N Hawking pairs are radiated theblack hole + radiation state will be | Ψ (cid:105) ≈ | Ψ (cid:105) M ⊗ | Ψ (cid:105) ⊗ | Ψ (cid:105) ⊗ . . . ⊗ | Ψ (cid:105) N , (1)where | Ψ (cid:105) i is the state of the i th Hawking pair. Since (1) has been written asa direct product of the black hole state | Ψ (cid:105) M and the Hawking quanta | Ψ (cid:105) i , theHawking radiation will in general not contain any information about the internalstate of the black hole system M .3. The stretching of spacelike slices will cause creation/annihalation operators on dif-ferent slices to be linearly related. The states will be related by [21, 22], | Ψ (cid:105) pair = Ce βc † b † | (cid:105) c | (cid:105) b . (2)4here β is a c -number and c † and b † are respectively, creation operators of the ingo-ing Hawking particle that is captured by the black hole, and the outgoing Hawkingparticle that blasts out of the black hole. Here | (cid:105) represents the vacuum state.To linear order, | Ψ (cid:105) pair = 1 √ | (cid:105) c | (cid:105) b + | (cid:105) c | (cid:105) b ) . (3)We will use this Bell pair state as an approximation of Hawking radiation. Theingoing and outgoing ( b, c ) particles are maximally entangled. The entanglemententropy of the subsystem of the outgoing or subsystem of the ingoing radiation, fora single pair state (3) is S ent = log 2, and for N pairs is S ent = N log 2This monotonically increasing entanglement entropy lies at the heart of the black holeinformation paradox. Any solution would presumably include a mechanism to stop thegrowth of the entanglement entropy and reduce it to zero.Backreaction and quantum gravity effects will likely modify the Hawking pair state (3).As long as the effective field theory picture of the horizon physics holds, these correctionsare expected to be small .Small corrections can however build up and lead to large departures. For example,consider perturbing the horizontally polarized photon pure state ρ = | →(cid:105)(cid:104)→ | andcreating an admixture of with density matrix ˆ ρ (cid:48) = (1 − (cid:15) ) | →(cid:105)(cid:104)→ | + (cid:15) | ↑(cid:105)(cid:104)↑ | , where (cid:15) (cid:28)
1. The entanglement entropy of of the perturbed state is small and is given by S ( ˆ ρ (cid:48) ) = − (cid:0) (1 − (cid:15) ) log (1 − (cid:15) ) + (cid:15) log (cid:15) (cid:1) , (4)The fidelity , given by F ( ρ , ρ ) = Tr (cid:113) ρ / ρ ρ / , measures the closeness of two states.There is high fidelity between ˆ ρ and ˆ ρ (cid:48) , since F ( ˆ ρ, ˆ ρ (cid:48) ) = √ − (cid:15) →
1. However, thefidelity between 2 N unperturbed photons described by ˆ ρ ⊗ N , and 2 N perturbed photonsdescribed by ˆ ρ (cid:48)⊗ N will become small as N becomes large. The two states look less andless like each other, since F (cid:0) ˆ ρ ⊗ N , ˆ ρ (cid:48)⊗ N (cid:1) = (1 − (cid:15) ) N/ → II. SMALL CORRECTIONS TO THE HAWKING STATE - MATHUR’S BOUND
Hawking particles cannot carry information out of the black hole if the Hawking pairsare always in the same state – such as in (3). We therefore consider a space of Hawkingstates, with dimensionality larger than one. Binary sequences are a popular way to encodeinformation. We therefore consider a two dimensional space of Hawking states spannedby V = { S (1) , S (2) } where for the ( n + 1)th emission S (1) = 1 √ (cid:16) | (cid:105) c n +1 | (cid:105) b n +1 + | (cid:105) c n +1 | (cid:105) b n +1 (cid:17) S (2) = 1 √ (cid:16) | (cid:105) c n +1 | (cid:105) b n +1 − | (cid:105) c n +1 | (cid:105) b n +1 (cid:17) . (5)A larger space such as V = { S (1) , S (2) , S (3) , S (4) } where S (3) = | (cid:105) c n +1 | (cid:105) b n +1 and S (4) = | (cid:105) c n +1 | (cid:105) b n +1 could have been chosen as was done in [11]. However, for our purposes,sequences from ( V ) ⊗ N possess enough complexity to encode the information in a blackhole.The complete system consists of the matter M inside a black hole, incoming Hawkingparticles c i , and outgoing Hawking particles b i . Given a basis | ψ i (cid:105) for the { M, c } systemof black hole matter and ingoing radiation, and a basis | χ i (cid:105) for the b i quanta, the fullstate upon diagonalization is | Ψ M,c , ψ b ( t n ) (cid:105) = (cid:88) i C i | ψ i (cid:105)| χ i (cid:105) . (6)At the next time-step of evolution, the b i quanta move a distance O ( M − ) from thehole. The next Hawking particle is emitted a time of order O ( M − ) after the previousemission. Thus, the b i and { b i +1 , c i +1 } systems will to a good approximation be causallydisconnected. We therefore assume that the b i particle is not affected by the ( i + 1)th,and later emissions.The ( i + 1)th emission will change the black hole states to | ψ i (cid:105) −→ S (1) | ψ (1) i (cid:105) + S (2) | ψ (2) i (cid:105) , where (cid:107)| ψ (1) i (cid:105)(cid:107) + (cid:107)| ψ (2) i (cid:105)(cid:107) = 1 . (7)In (7) the Hawking pair state is entangled with the black hole, in contrast to (1). Thisenables correlations between the Hawking quanta, the black hole state, and previouslyemitted Hawking c quanta to exist. It has been hoped that such correlations will allowinformation to be carried out by Hawking quanta. (In the leading order analysis a new6awking pair is in the state S (1) such that | ψ (1) i (cid:105) = | ψ i (cid:105) and | ψ (2) i (cid:105) = 0, and there are nocorrelations.)The complete state after the ( n + 1)th emission is, | Ψ M,c , ψ b ( t n +1 ) (cid:105) = (cid:88) i C i (cid:104) S (1) | ψ (1) i (cid:105) + S (2) | ψ (2) i (cid:105) (cid:105) | χ i (cid:105) = Λ (1) S (1) + Λ (2) S (2) , (8)where, Λ (1) = (cid:88) i C i | ψ (1) i (cid:105)| χ i (cid:105) ; Λ (2) = (cid:88) i C i | ψ (2) i (cid:105)| χ i (cid:105) . (9)In (8) we have tensored with | χ i (cid:105) , the state of the previously emitted b quanta. This isbecause we have assumed that new b emissions don’t affect previously emitted b quanta.The { b , . . . , b n } quanta are not affected by later emissions. Thus, their entanglemententropy, S ( { b , . . . , b n } ) = (cid:80) i | C i | log | C i | , stays the same at time-step t n +1 .The entanglement entropy of the ( b n +1 , c n +1 ) pair with the rest of the system is givenby the density matrix of the ( b n +1 , c n +1 ) system,ˆ ρ b n +1 ,c n +1 = (cid:104) Λ (1) | Λ (1) (cid:105) (cid:104) Λ (1) | Λ (2) (cid:105)(cid:104) Λ (2) | Λ (1) (cid:105) (cid:104) Λ (2) | Λ (2) (cid:105) where (cid:107) Λ (1) (cid:107) + (cid:107) Λ (2) (cid:107) = 1 (10)Mathur defined a modification to the Hawking state to be ‘small’ if [10] (cid:107) Λ (2) (cid:107) < (cid:15) where (cid:15) (cid:28) . (11)In this framework of small corrections, Hawking pairs are predominantly produced in the S (1) state and are rarely produced in the S (2) state. Mathur showed that the entanglemententropy at each time-step increases by at least log 2 − (cid:15) , which is positive for small (cid:15) ,[10]. Models with correlations between consecutive emissions [11] and other toy modelsas in [12], have echoed the inability of small corrections in decreasing the entanglemententropy increase at each time step.In the next section we shall illustrate Mathur’s bound in a toy model that incorporatesa more general correlation compared to the model presented in [11]. IV. A SIMPLE TOY MODEL WITH ARBITRARY CORRECTIONS
We now consider a model that allows all previously radiated quanta to modify thestate of a new Hawking pair. Let the first Hawking pair emitted be a Bell pair state | Ψ (cid:105) = 1 √ | (cid:105) b | (cid:105) c + | (cid:105) b | (cid:105) c ) (12)7e assume that the Hilbert space of the newly evolved pair is spanned by {| (cid:105) , | (cid:105)} .The n pair state is given by | Ψ n (cid:105) = n − (cid:88) i =0 a i | i (cid:105) b | i (cid:105) c where n − (cid:88) i =0 | a i | = 1 (13)Here | i (cid:105) c and | i (cid:105) b denote the states of the n ingoing and n outgoing quanta respectively,and i is the n bit representation of the integer i . We build the state of the ( n + 1)th pairfrom the previous n quanta and the ( n + 1)th quanta as the entangled state, | Ψ n +1 (cid:105) = n − (cid:88) i =0 a i | i (cid:105) b | i (cid:105) c ⊗ √ (cid:16) e s i,n, | (cid:105) b | (cid:105) c + e s i,n, | (cid:105) b | (cid:105) c (cid:17) (14)The term √ e s i,n,j is the amplitude to observe the new pair in the state | j (cid:105) b | j (cid:105) c , giventhat the earlier pairs were given by | i (cid:105) b | i (cid:105) c . If the modification to the Bell state for the( n + 1)th pair is small, then | s i,n,j | is a small positive number. Normalization requires, (cid:88) j =0 e s i,n,j = 2 . (15)The entanglement entropy of the n + 1 radiated quanta is hence given by (where log istaken have base e ), S ( n + 1) = − (cid:88) i (cid:88) j (cid:18) a i e s i,n,j √ (cid:19) log (cid:18) a i e s i,n,j √ (cid:19) = − (cid:88) i (cid:88) j a i e s i,n,j (cid:18) log a i + s i,n,j −
12 log 2 (cid:19) = − (cid:88) i a i log a i + log 2 − (cid:88) i a i (cid:88) j s i,n,j e s i,n,j . (16)The first term in (16) is the entanglement entropy of the first n quanta. Upon defining∆ S = S ( n + 1) − S ( n ), we find that∆ S = log 2 − (cid:88) i a i (cid:88) j s i,n,j e s i,n,j . (17)We can find an upper and lower bound on ∆ S . Consider the quantity f i,n = (cid:88) j s i,n,j e s i,n,j = s i,n, e s i,n, + s i,n, e s i,n, , (18)where s i,n, , s i,n, ∈ R . Using Lagrange Multipliers we can easily see that f i,n ≥ s i,n, = s i,n, = 0. The normalization (15) requires that s i,n, ≤ log 2 and s i,n, ≤ log 2. Therefore, using (15) again, we find that f i,n ≤ · log 2 = log 2.8e therefore find that, 0 ≤ ∆ S ≤ log 2 . (19)Our calculation therefore gives a lower bound to ∆ S . This improves Mathur’s lowerbound of ∆ S ≥ log 2 − (cid:15) which is positive only if (cid:15) is small. Our calculation shows thateven large corrections in this model cannot lead to entropy decrease. This leads us to themore general model in the next section. V. GENERALIZATION OF MATHUR’S BOUND
In this section we generalize Mathur’s bound. Some of this work was presented ina preliminary form in [26]. We allow corrections of arbitrary magnitude to the leadingorder analysis. Although, arbitrary corrections may destroy the
Solar System Limit asdescribed by Mathur, and expectations of no-drama at the horizon, our results establishinteresting and nontrivial strong upper and lower bounds on ∆ S , which must be obeyedeven if (cid:15) ∼ (cid:104) Λ (2) | Λ (2) (cid:105) = (cid:15) , (cid:104) Λ (1) | Λ (1) (cid:105) = 1 − (cid:15) (cid:104) Λ (1) | Λ (2) (cid:105) = (cid:104) Λ (2) | Λ (1) (cid:105) = (cid:15) , γ = 1 − (cid:16) (cid:15) (1 − (cid:15) ) − (cid:15) (cid:17) . (20)We define a correction to be small if | (cid:15) | (cid:28)
1. Although (cid:15) is generally complex and (cid:104) Λ (1) | Λ (2) (cid:105) = (cid:104) Λ (2) | Λ (1) (cid:105) ∗ , for simplicity in (20) we assume (cid:15) to be real. Complexifying itgives the same result. Lemma 1.
The entanglement entropy of the newly created pair is given by S ( p ) ≤ (cid:112) − γ log 2 . Proof.
The reduced density matrix for the pair isˆ ρ p = − (cid:15) (cid:15) (cid:15) (cid:15) . (21)The eigenvalues of this matrix are: λ = γ and λ = − γ , where γ = (cid:112) − (cid:15) (1 − (cid:15) ) − (cid:15) ).Thus 0 ≤ γ ≤
1. This implies that0 ≤ (cid:15) (1 − (cid:15) ) − (cid:15) ≤ . (22)9ence, the entanglement entropy of the pair is S ( p ) = − tr ˆ ρ p log ˆ ρ p = − (cid:88) i =1 λ i log λ i = log 2 −
12 [(1 + γ ) log(1 + γ ) + (1 − γ ) log(1 − γ )] . (23)It can be shown that for 0 ≤ x ≤ − x ) log 2 ≤ log 2 −
12 [(1 + x ) log(1 + x ) + (1 − x ) log(1 − x )] ≤ √ − x log 2 . (24)The result follows from (24). Lemma 2. (1 − (cid:15) ) log 2 ≤ S ( b n +1 ) = S ( c n +1 ) ≤ (cid:113) − (cid:15) log 2 Proof.
The complete state of the system after the creation of n + 1 pairs is | Ψ M,c , ψ b ( t n +1 ) (cid:105) = (cid:104) | (cid:105) c n +1 | (cid:105) b n +1 √ (1) + Λ (2) ) (cid:105) + (cid:104) | (cid:105) c n +1 | (cid:105) b n +1 √ (1) − Λ (2) ) (cid:105) , (25)where Λ (1) and Λ (2) reflect the state of the black hole and are defined by (9).Now, the reduced density matrix of the c n +1 or b n +1 quanta isˆ ρ b n +1 = ˆ ρ c n +1 = (cid:104) (Λ (1) + Λ (2) ) | (Λ (1) + Λ (2) ) (cid:105) (cid:104) (Λ (1) − Λ (2) ) | (Λ (1) − Λ (2) ) (cid:105) = (cid:15) − (cid:15) . (26)Then, entanglement entropy of the c n +1 or b n +1 quanta is S ( b n +1 ) = S ( c n +1 ) = log 2 − (cid:15) (cid:15) ) − − (cid:15) − (cid:15) ) . (27)The result follows directly from (24).Using these lemmas we now prove the following theorem.10 heorem 1. The change in the entanglement entropy from time-step t n to t n +1 is re-stricted by the following bound: − (cid:15) − (cid:112) − γ ≤ ∆ S log 2 ≤ (cid:113) − (cid:15) (28) where ∆ S = S ( b n +1 , { b } ) − S ( { b } ) .Proof. Let us assume A = { b } and B = b n +1 and C = c n +1 . Using the strong subadditivityinequality , S ( A ) + S ( C ) ≤ S ( A, B ) + S ( B, C ) we find that, S ( { b } ) + S ( c n +1 ) ≤ S ( { b } , b n +1 ) + S ( b n +1 , c n +1 ) ⇒ ∆ S ≥ (1 − (cid:15) ) log 2 − (cid:112) − γ log 2 . (29)Inequality (29) follows from using Lemma 1 and Lemma 2.Now using the subadditivity inequality , S ( A ) + S ( B ) ≥ S ( A, B ), we find, S ( { b } ) + S ( b n +1 ) ≥ S ( { b } , b n +1 ) ⇒ ∆ S ≤ (cid:113) − (cid:15) log 2 (30)Inequality (30) follows from Lemma 2 and combining (29) and (30) gives Theorem 1.The change in the entanglement entropy upper and lower bounds in Theorem 1 holdirrespective of the size of the correction (cid:15) . They hold in particular for O (1) corrections. Inthe given effective field theory framework, they thus give an upper bound for the changein entanglement entropy for non-trivial corrections and can help us understand the sizeof the correction needed to restore unitarity. A. Nontriviality
When (cid:15) →
0, the Hawking radiation is always in the same state, S (1) . Thus theHawking radiation can’t carry out information from the black hole. This is reflected bythe upper and lower bounds in (28) being the same and thus the entropy increase ismaximal. The case of (cid:15) → S (2) and the entropy increase given by (28) is thus maximal. Thus (28) givesa non-trivial bound for (cid:15) (cid:54) = 0 and (cid:15) (cid:54) = 1, particularly for large corrections. The analysis11n [10] and the various generalizations reviewed in [14] do not address the case of largecorrections.Consider the maximum difference between the lower and upper bound in (28). Denot-ing this quantity by D ∆ S , we find D ∆ S = log 2 (cid:18)(cid:113) − (cid:15) + (cid:113) (cid:15) (1 − (cid:15) ) − (cid:15) ) − (1 − (cid:15) ) (cid:19) (31)It is straightforward to show that for any value of (cid:15) , D ∆ S is maximized when (cid:15) (1 − (cid:15) )is maximal, i.e. . Then (31) reduces to D ∆ S log 2 = 2 (cid:113) − (cid:15) − (1 − (cid:15) ) (32)It is straightforward to show that maximization of D ∆ S requires dD ∆ S d(cid:15) = 0 ⇒ (cid:15) = 0.This can also be seen from (22). This gives, D ∆ S log 2 ≤ . (33)Thus D ∆ S never exceeds log 2. While it is clear that that ∆ S ≤ log 2, the result that D ∆ S ≤ log 2 is a different nontrivial bound. B. Small corrections are not enough
In the beginning of the black hole evaporation process the entanglement entropy ofthe outgoing pairs will be significant because the outgoing radiation will carry little infor-mation about the hole. If the evaporation is unitary then as more Hawking particles areemitted, more and more information about the hole will come out and the entanglemententropy will decrease [17]. Once all of the information has come out, the entanglemententropy will be zero as the { b } system will be a pure state.This means that at some point the lower bound in (28) must become negative. Theoccurs when, 4 (cid:15) (1 − (cid:15) ) > − (cid:15) (1 − (cid:15) ) . (34)The maximum of the left hand side of (34) is . This implies that1 − (cid:15) (1 − (cid:15) ) <
14 = ⇒ < (cid:15) < √ . (35)12his gives a necessary (but not sufficient) condition for unitarity. Equation (35) impliesthat relatively large corrections are needed. The bound in (28) does not require the cor-rection parameters to be small and thus it remains of interest even in the large correctionregime given by (35). This is a unique feature of the derivation presented in this paperand will contribute to the results in the following section. VI. INCOMPATIBILITY OF BELL PAIR STATE CORRECTION TO PAGECURVES
In [17], Page showed that the a small subsystem like the outgoing radiation emittedby a black hole at the beginning of evaporation [18], is maximally entangled with thelarger system. Mathematical proofs were provided in [27–29].Suppose the size of the smaller subsystem’s Hilbert space is m and the larger subsys-tem’s Hilbert space dimension is n , such that the combined system has dimension mn .Suppose the combined system is described by a random pure state. Page showed that inthis case the entropy S m,n , of the smaller subsystem is S m,n = log m − m n + O (cid:0) ( m/n ) (cid:1) (36)Thus for n (cid:29) m , S m,n is the very near the course-grained entropy of the outgoing radia-tion, which is log m . Also, the outgoing radiation is thermal .As the outgoing radiation entropy rises, the size of the outgoing radiation subsystem m , increases. This means that information will leak out of the black hole with increasingtime. At some point enough information will have leaked out such that m ≈ n and S m.n will decrease. Eventually, we will have m (cid:29) n . The analysis will then be similar to the n (cid:29) m case. Broadly speaking, the m (cid:29) n and n (cid:29) m cases describe similar systems,but with m and n interchanged. The fully symmetric case for the Page Curve is shownin Figure 1a. Note, the turnover point, or Page Time, occurs when half of the black holehas evaporated.However, in general the Page Curve is not symmetric because the entropy increaseof the outgoing radiation, dS b /dt is not equal to the energy decrease of the black hole, − dS BH /dt . This is due to the different greybody factors, helicities and particle numbersavailable for massless emission that affect the black hole entropy loss and Hawking radi-ation entropy differently. The corresponding Page Curve is shown in Figure 1b. In [19],13 ) Page Curve b) Includes black hole physics c) Page-like curvesFIG. 1. using previous scattering calculations, Page showed that β ≡ dS b /dt − dS BH /dt ≈ .
484 72He also showed that the Page Time, t P age is related to the black hole lifetime t decay , as t P age = (cid:34) − (cid:18) ββ + 1 (cid:19) / (cid:35) t decay ≈ . t decay for β = 1 .
484 72 . Thus, the Page Time is heavily dependent on β .Our result in (28) generates the envelope of any Page-like curve for the evaporation viaBell pair emission model that we have been considering. The monotonically increasingpart is bounded by a straight line with a slope equal to the maximum of the upper boundin (28). This is equal to log 2. Similarly, the decreasing part is also bounded by a straightline and the slope of this line is obtained from the minimum of the lower bound in (28).The lower bound = 1 − (cid:15) − (cid:112) − γ = 1 − (cid:15) − (cid:113) (cid:15) (1 − (cid:15) ) − (cid:15) . For any value of (cid:15) , this quantity is minimized when (cid:15) (1 − (cid:15) ) is maximized. This occurswhen (cid:15) (1 − (cid:15) ) = . Therefore, we need to consider the minimum value of the quantity1 − (cid:15) − (cid:112) − (cid:15) , which is − . The Page-like curve that our evaporation model generatesis bounded by the grey region in Figure 1c. Note, in drawing the graph in Figure 1c, wefirst fixed the evaporation time t decay . Then we imposed the two bounds.We will take the intersection of both bounds in Figure 1c to be the Page Time in thismodel. This occurs at t P age = 0 . t decay and corresponds to β = 6 .
24. This correspondsto β = 6 .
24 Such an early Page Time would imply that information quickly comes out of14he black hole because the emitted photons/gravitons in the outgoing radiation possessmuch more entropy – by a factor of 6 .
24 – than the entropy decrease of the black hole.This seems farfetched and would seem to indicate that evaporation via only Bell pairstates doesn’t correspond to the physical picture of evaporation in Page’s calculations.It also implies that maximal entanglement between the outgoing radiation and the holedoesn’t last very long.This picture is at odds with Page’s description of unitary black hole evaporation. InPage’s description the Page Time is near the half-way point in the evaporation process.General arguments regarding subsystem entropy transfer support the belief that t P age ≈ t decay and reasonable estimates of β also support this. Adami and Bradler also foundmodifications to the Page Curve in their study of black hole dynamics using a trilinearHamiltonian [30]. In retrospect, the early turnover of the envelope of our Page Curvemay not be surprising because a non-negligible (cid:15) and (cid:15) will presumably create strongerinteractions and cause the Page Curve’s envelope to turn over more quickly.Giddings and Shi [31] generalized Mathur’s model and showed that corrections viaBell pair states only, would not restore unitarity. Our Page Curve puts contraints onBell pair emission that are unlikely to be met and thus reaffirms Giddings and Shi’s, andMathur’s earlier claim that Bell pairs don’t restore unitarity. VII. CONCLUSION
Mathur’s analysis of ‘small’ corrections to the leading order Hawking analysis showedthat Bell Pair states do not unitarize the black hole evaporation process. In this paper,we reaffirm this claim by analyzing a toy qubit model that incorporates a generalizedquantum correlation between successive pairs. We then generalize Mathur’s work byrelaxing the ‘smallness’ condition. We establish a rigorous nontrivial upper and lowerbound for the change in entanglement entropy. This enables us to parameterize therequired correction to the Hawking state. Our results show that the correction requiredto restore unitarity is ‘not-so-small’ or even ‘large’. However, even if we allow such anevaporation law, we find that it is at odds with the expected Page Curve. If a black hole’stime evolution is to be unitary, the entropy of its outgoing radiation should approximatethe expected Page curve. Thus, this leads us to reaffirm the belief that the information15aradox cannot be resolved by adhering to the picture of an evaporation process via onlyBell Pair states.
ACKNOWLEDGEMENTS
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